Generalized Functions; Multiplication of Distributions; Applications to Elasticity, Elastoplasticity, Fluid Dynamics and Acoustics

If Ω denotes any open set in ℝn, I have defined an algebra G (Ω) of “generalized functions” on Ω. One has the set of inclusions $${\text{C}}^\infty \left( \Omega \right) \subset D\prime \left( \Omega \right) \subset G\left( \Omega \right)$$ where C∞ (Ω) (respectively D’ (Ω) denotes the set of all C∞ functions (resp. all distributions) on Ω. Two basic points have to be stressed: C∞ (Ω), with its usual pointwise multiplication, is a subalgebra of G(Ω)any element of G(Ω) admits partial derivatives of any order which generalize exactly those in D’ (Ω).